Question

Simplify: ${\tan }^{-1}\left(\frac{\sin x}{1+\cos x}\right)$

Answer 1

Consider the following question.

${\tan }^{-1}\left(\frac{\sin x}{1+\cos x}\right)$

$={\tan }^{-1}\left(\frac{2\sin \frac{x}{2}\cos \frac{x}{2}}{2{\cos }^2\frac{x}{2}}\right)$

$={\tan }^{-1}\left(\tan \frac{x}{2}\right)$

$=\frac{x}{2}$

Hence, this is the required answer.